|
# |
ODE |
Program classification |
CAS classification |
Solved? |
Verified? |
time (sec) |
|
\[ {}{y^{\prime }}^{4} = \left (y-a \right )^{3} \left (y-b \right )^{2} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
4.075 |
|
|
\[ {}{y^{\prime }}^{4}+f \left (x \right ) \left (y-a \right )^{3} \left (y-b \right )^{2} = 0 \] |
first_order_nonlinear_p_but_separable |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
4.665 |
|
|
\[ {}{y^{\prime }}^{4}+f \left (x \right ) \left (y-a \right )^{3} \left (y-b \right )^{3} = 0 \] |
first_order_nonlinear_p_but_separable |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
12.732 |
|
|
\[ {}{y^{\prime }}^{4}+f \left (x \right ) \left (y-a \right )^{3} \left (y-b \right )^{3} \left (y-c \right )^{2} = 0 \] |
first_order_nonlinear_p_but_separable |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
668.064 |
|
|
\[ {}{y^{\prime }}^{4}+x y^{\prime }-3 y = 0 \] |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
1.38 |
|
|
\[ {}{y^{\prime }}^{4}-4 x^{2} y {y^{\prime }}^{2}+16 x y^{2} y^{\prime }-16 y^{3} = 0 \] |
unknown |
[[_homogeneous, ‘class G‘]] |
✗ |
N/A |
1.309 |
|
|
\[ {}{y^{\prime }}^{4}+4 y {y^{\prime }}^{3}+6 y^{2} {y^{\prime }}^{2}-\left (1-4 y^{3}\right ) y^{\prime }-\left (3-y^{3}\right ) y = 0 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
1.856 |
|
|
\[ {}2 {y^{\prime }}^{4}-y y^{\prime }-2 = 0 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.622 |
|
|
\[ {}x {y^{\prime }}^{4}-2 y {y^{\prime }}^{3}+12 x^{3} = 0 \] |
unknown |
[[_1st_order, _with_linear_symmetries]] |
✗ |
N/A |
2.131 |
|
|
\[ {}3 {y^{\prime }}^{5}-y y^{\prime }+1 = 0 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.233 |
|
|
\[ {}{y^{\prime }}^{6} = \left (y-a \right )^{4} \left (y-b \right )^{3} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
91.425 |
|
|
\[ {}{y^{\prime }}^{6}+f \left (x \right ) \left (y-a \right )^{4} \left (y-b \right )^{3} = 0 \] |
first_order_nonlinear_p_but_separable |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
99.567 |
|
|
\[ {}{y^{\prime }}^{6}+f \left (x \right ) \left (y-a \right )^{5} \left (y-b \right )^{3} = 0 \] |
first_order_nonlinear_p_but_separable |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
213.078 |
|
|
\[ {}{y^{\prime }}^{6}+f \left (x \right ) \left (y-a \right )^{5} \left (y-b \right )^{4} = 0 \] |
first_order_nonlinear_p_but_separable |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
778.25 |
|
|
\[ {}x^{2} \left ({y^{\prime }}^{6}+3 y^{4}+3 y^{2}+1\right ) = a^{2} \] |
unknown |
[_rational] |
❇ |
N/A |
15.077 |
|
|
\[ {}2 \sqrt {a y^{\prime }}+x y^{\prime }-y = 0 \] |
clairaut |
[[_homogeneous, ‘class G‘], _Clairaut] |
✓ |
✓ |
2.11 |
|
|
\[ {}\left (x -y\right ) \sqrt {y^{\prime }} = a \left (1+y^{\prime }\right ) \] |
dAlembert |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
3.405 |
|
|
\[ {}2 \left (y+1\right )^{\frac {3}{2}}+3 x y^{\prime }-3 y = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
8.423 |
|
|
\[ {}\sqrt {1+{y^{\prime }}^{2}}+a y^{\prime } = x \] |
quadrature |
[_quadrature] |
✓ |
✓ |
2.527 |
|
|
\[ {}\sqrt {1+{y^{\prime }}^{2}}+a y^{\prime } = y \] |
quadrature |
[_quadrature] |
✓ |
✓ |
2.367 |
|
|
\[ {}\sqrt {1+{y^{\prime }}^{2}} = x y^{\prime } \] |
quadrature |
[_quadrature] |
✓ |
✓ |
1.731 |
|
|
\[ {}\sqrt {a^{2}+b^{2} {y^{\prime }}^{2}}+x y^{\prime }-y = 0 \] |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
4.38 |
|
|
\[ {}a \sqrt {1+{y^{\prime }}^{2}}+x y^{\prime }-y = 0 \] |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
2.182 |
|
|
\[ {}a x \sqrt {1+{y^{\prime }}^{2}}+x y^{\prime }-y = 0 \] |
dAlembert |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
0.764 |
|
|
\[ {}\sqrt {\left (x^{2} a +y^{2}\right ) \left (1+{y^{\prime }}^{2}\right )}-y y^{\prime }-x a = 0 \] |
dAlembert |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
246.119 |
|
|
\[ {}a \left (1+{y^{\prime }}^{3}\right )^{\frac {1}{3}}+x y^{\prime }-y = 0 \] |
clairaut |
[_Clairaut] |
✓ |
✓ |
183.375 |
|
|
\[ {}\cos \left (y^{\prime }\right )+x y^{\prime } = y \] |
clairaut |
[_Clairaut] |
✓ |
✓ |
0.425 |
|
|
\[ {}a \cos \left (y^{\prime }\right )+b y^{\prime }+x = 0 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.26 |
|
|
\[ {}\sin \left (y^{\prime }\right )+y^{\prime } = x \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.269 |
|
|
\[ {}y^{\prime } \sin \left (y^{\prime }\right )+\cos \left (y^{\prime }\right ) = y \] |
quadrature |
[_quadrature] |
✓ |
✓ |
1.821 |
|
|
\[ {}{y^{\prime }}^{2} \left (x +\sin \left (y^{\prime }\right )\right ) = y \] |
dAlembert |
[_dAlembert] |
✓ |
✓ |
0.898 |
|
|
\[ {}\left (1+{y^{\prime }}^{2}\right ) \sin \left (-y+x y^{\prime }\right )^{2} = 1 \] |
clairaut |
[_Clairaut] |
✓ |
✓ |
3.925 |
|
|
\[ {}\left (1+{y^{\prime }}^{2}\right ) \left (\arctan \left (y^{\prime }\right )+x a \right )+y^{\prime } = 0 \] |
quadrature |
[_quadrature] |
✓ |
✗ |
1.71 |
|
|
\[ {}{\mathrm e}^{y^{\prime }-y}-{y^{\prime }}^{2}+1 = 0 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.203 |
|
|
\[ {}\ln \left (y^{\prime }\right )+x y^{\prime }+a = 0 \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.567 |
|
|
\[ {}\ln \left (y^{\prime }\right )+x y^{\prime }+a = y \] |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.879 |
|
|
\[ {}\ln \left (y^{\prime }\right )+x y^{\prime }+a +b y = 0 \] |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
3.114 |
|
|
\[ {}\ln \left (y^{\prime }\right )+4 x y^{\prime }-2 y = 0 \] |
dAlembert |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
1.312 |
|
|
\[ {}\ln \left (y^{\prime }\right )+a \left (-y+x y^{\prime }\right ) = 0 \] |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.828 |
|
|
\[ {}a \left (\ln \left (y^{\prime }\right )-y^{\prime }\right )-x +y = 0 \] |
dAlembert |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
1.364 |
|
|
\[ {}y \ln \left (y^{\prime }\right )+y^{\prime }-y \ln \left (y\right )-x y = 0 \] |
separable, homogeneousTypeD2 |
[_separable] |
✓ |
✓ |
3.098 |
|
|
\[ {}y^{\prime } \ln \left (y^{\prime }\right )-\left (1+x \right ) y^{\prime }+y = 0 \] |
clairaut |
[[_1st_order, _with_linear_symmetries], _Clairaut] |
✓ |
✓ |
0.868 |
|
|
\[ {}y^{\prime } \ln \left (y^{\prime }+\sqrt {a +{y^{\prime }}^{2}}\right )-\sqrt {1+{y^{\prime }}^{2}}-x y^{\prime }+y = 0 \] |
clairaut |
[_Clairaut] |
✓ |
✓ |
7.899 |
|
|
\[ {}\ln \left (\cos \left (y^{\prime }\right )\right )+y^{\prime } \tan \left (y^{\prime }\right ) = y \] |
unknown |
[_dAlembert] |
✗ |
N/A |
1.551 |
|
|
\[ {}y^{\prime } = \frac {x y}{x^{2}-y^{2}} \] |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.12 |
|
|
\[ {}y^{\prime } = \frac {x +y-3}{x -y-1} \] |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.603 |
|
|
\[ {}y^{\prime } = \frac {2 x +y-1}{4 x +2 y+5} \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.064 |
|
|
\[ {}y^{\prime }-\frac {2 y}{1+x} = \left (1+x \right )^{2} \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.708 |
|
|
\[ {}y^{\prime }+x y = x^{3} y^{3} \] |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
0.822 |
|
|
\[ {}\frac {2 x}{y^{3}}+\frac {\left (y^{2}-3 x^{2}\right ) y^{\prime }}{y^{4}} = 0 \] |
exact, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
✓ |
5.584 |
|
|
\[ {}y+x y^{2}-x y^{\prime } = 0 \] |
riccati, bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
✓ |
0.744 |
|
|
\[ {}y^{2} \left (1+{y^{\prime }}^{2}\right ) = R^{2} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.811 |
|
|
\[ {}y = x y^{\prime }+\frac {a y^{\prime }}{\sqrt {1+{y^{\prime }}^{2}}} \] |
clairaut |
[_Clairaut] |
✓ |
✓ |
60.916 |
|
|
\[ {}y = x {y^{\prime }}^{2}+{y^{\prime }}^{2} \] |
dAlembert |
[[_homogeneous, ‘class C‘], _rational, _dAlembert] |
✓ |
✓ |
0.309 |
|
|
\[ {}\left (1+x \right ) y+\left (1-y\right ) x y^{\prime } = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.963 |
|
|
\[ {}y^{2}+x y^{2}+\left (x^{2}-x^{2} y\right ) y^{\prime } = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.661 |
|
|
\[ {}x y \left (x^{2}+1\right ) y^{\prime }-1-y^{2} = 0 \] |
exact, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
6.803 |
|
|
\[ {}1+y^{2}-\left (y+\sqrt {1+y^{2}}\right ) \left (x^{2}+1\right )^{\frac {3}{2}} y^{\prime } = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
23.264 |
|
|
\[ {}\sin \left (x \right ) \cos \left (y\right )-\cos \left (x \right ) \sin \left (y\right ) y^{\prime } = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.344 |
|
|
\[ {}\sec \left (x \right )^{2} \tan \left (y\right )+\sec \left (y\right )^{2} \tan \left (x \right ) y^{\prime } = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
40.663 |
|
|
\[ {}\left (y-x \right ) y^{\prime }+y = 0 \] |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.105 |
|
|
\[ {}\left (2 \sqrt {x y}-x \right ) y^{\prime }+y = 0 \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
3.198 |
|
|
\[ {}x y^{\prime }-y-\sqrt {x^{2}+y^{2}} = 0 \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
3.44 |
|
|
\[ {}x -y \cos \left (\frac {y}{x}\right )+x \cos \left (\frac {y}{x}\right ) y^{\prime } = 0 \] |
homogeneousTypeD, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.442 |
|
|
\[ {}8 y+10 x +\left (5 y+7 x \right ) y^{\prime } = 0 \] |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.527 |
|
|
\[ {}2 x -y+1+\left (2 y-1\right ) y^{\prime } = 0 \] |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
3.587 |
|
|
\[ {}3 y-7 x +7+\left (7 y-3 x +3\right ) y^{\prime } = 0 \] |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
3.013 |
|
|
\[ {}y^{\prime }+\frac {x y}{x^{2}+1} = \frac {1}{2 \left (x^{2}+1\right ) x} \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.835 |
|
|
\[ {}x \left (-x^{2}+1\right ) y^{\prime }+\left (2 x^{2}-1\right ) y = a \,x^{3} \] |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.347 |
|
|
\[ {}y^{\prime }+\frac {y}{\left (-x^{2}+1\right )^{\frac {3}{2}}} = \frac {x +\sqrt {-x^{2}+1}}{\left (-x^{2}+1\right )^{2}} \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
4.732 |
|
|
\[ {}y^{\prime }+y \cos \left (x \right ) = \frac {\sin \left (2 x \right )}{2} \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.525 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime }+y = \arctan \left (x \right ) \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.905 |
|
|
\[ {}\left (-x^{2}+1\right ) z^{\prime }-x z = a x z^{2} \] |
exact, riccati, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
3.239 |
|
|
\[ {}3 z^{2} z^{\prime }-a z^{3} = 1+x \] |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_rational, _Bernoulli] |
✓ |
✓ |
1.86 |
|
|
\[ {}z^{\prime }+2 x z = 2 a \,x^{3} z^{3} \] |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
0.997 |
|
|
\[ {}z^{\prime }+z \cos \left (x \right ) = z^{n} \sin \left (2 x \right ) \] |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
32.015 |
|
|
\[ {}x y^{\prime }+y = y^{2} \ln \left (x \right ) \] |
riccati, bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
1.004 |
|
|
\[ {}x^{3}+3 x y^{2}+\left (y^{3}+3 x^{2} y\right ) y^{\prime } = 0 \] |
exact |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
✓ |
0.334 |
|
|
\[ {}1+\frac {y^{2}}{x^{2}}-\frac {2 y y^{\prime }}{x} = 0 \] |
exact |
[[_homogeneous, ‘class A‘], _exact, _rational, _Bernoulli] |
✓ |
✓ |
0.256 |
|
|
\[ {}\frac {3 x}{y^{3}}+\left (\frac {1}{y^{2}}-\frac {3 x^{2}}{y^{4}}\right ) y^{\prime } = 0 \] |
exact |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.381 |
|
|
\[ {}x +y y^{\prime }+\frac {x y^{\prime }}{x^{2}+y^{2}}-\frac {y}{x^{2}+y^{2}} = 0 \] |
exact |
[[_1st_order, _with_linear_symmetries], _exact, _rational] |
✓ |
✓ |
0.434 |
|
|
\[ {}1+{\mathrm e}^{\frac {x}{y}}+{\mathrm e}^{\frac {x}{y}} \left (1-\frac {x}{y}\right ) y^{\prime } = 0 \] |
exact |
[[_homogeneous, ‘class A‘], _exact, _dAlembert] |
✓ |
✓ |
0.387 |
|
|
\[ {}{\mathrm e}^{x} \left (x^{2}+y^{2}+2 x \right )+2 y \,{\mathrm e}^{x} y^{\prime } = 0 \] |
exact |
[[_homogeneous, ‘class D‘], _exact, _rational, _Bernoulli] |
✓ |
✓ |
0.313 |
|
|
\[ {}n \cos \left (n x +m y\right )-m \sin \left (m x +n y\right )+\left (m \cos \left (n x +m y\right )-n \sin \left (m x +n y\right )\right ) y^{\prime } = 0 \] |
exact |
[_exact] |
✓ |
✓ |
0.605 |
|
|
\[ {}\frac {x}{\sqrt {1+x^{2}+y^{2}}}+\frac {y y^{\prime }}{\sqrt {1+x^{2}+y^{2}}}+\frac {y}{x^{2}+y^{2}}-\frac {x y^{\prime }}{x^{2}+y^{2}} = 0 \] |
exact |
[[_1st_order, _with_linear_symmetries], _exact] |
✓ |
✓ |
0.865 |
|
|
\[ {}\frac {x^{n} y^{\prime }}{b y^{2}-c \,x^{2 a}}-\frac {a y x^{a -1}}{b y^{2}-c \,x^{2 a}}+x^{a -1} = 0 \] |
riccati |
[_Riccati] |
✓ |
✓ |
40.685 |
|
|
\[ {}2 x y+\left (y^{2}-2 x^{2}\right ) y^{\prime } = 0 \] |
exact |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.365 |
|
|
\[ {}\frac {1}{x}+\frac {y^{\prime }}{y}+\frac {2}{y}-\frac {2 y^{\prime }}{x} = 0 \] |
exact |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.227 |
|
|
\[ {}-y+x y^{\prime } = \sqrt {x^{2}+y^{2}} \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
2.613 |
|
|
\[ {}8 y+10 x +\left (5 y+7 x \right ) y^{\prime } = 0 \] |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.67 |
|
|
\[ {}x^{2}+2 x y-y^{2}+\left (y^{2}+2 x y-x^{2}\right ) y^{\prime } = 0 \] |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
2.637 |
|
|
\[ {}y^{2}+\left (x y+x^{2}\right ) y^{\prime } = 0 \] |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
2.676 |
|
|
\[ {}\left (x \cos \left (\frac {y}{x}\right )+y \sin \left (\frac {y}{x}\right )\right ) y+\left (x \cos \left (\frac {y}{x}\right )-y \sin \left (\frac {y}{x}\right )\right ) x y^{\prime } = 0 \] |
exact |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
0.647 |
|
|
\[ {}\left (x^{2} y^{2}+x y\right ) y+\left (x^{2} y^{2}-1\right ) x y^{\prime } = 0 \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.265 |
|
|
\[ {}\left (x^{3} y^{3}+x^{2} y^{2}+x y+1\right ) y+\left (x^{3} y^{3}-x^{2} y^{2}-x y+1\right ) x y^{\prime } = 0 \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
1.514 |
|
|
\[ {}x^{2}+y^{2}+2 x +2 y y^{\prime } = 0 \] |
exact |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
✓ |
0.229 |
|
|
\[ {}x^{2}+y^{2}-2 x y y^{\prime } = 0 \] |
exact |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
0.204 |
|
|
\[ {}2 x y+\left (y^{2}-3 x^{2}\right ) y^{\prime } = 0 \] |
exact |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.243 |
|
|
\[ {}y+\left (2 y-x \right ) y^{\prime } = 0 \] |
exact |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.33 |
|
|
\[ {}x y^{\prime }-a y+y^{2} = x^{-2 a} \] |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
0.6 |
|
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